Question

dx/dt=2y-x
dy/dt=e^x+y

solve y=f(x)

Answer 1

can we just use the fact that it is exact?

Answer 2

Hm ... are you sure the second equation is e^x not e^t?

Answer 3

Let me start by substituting \(2y=x'+x \implies 2y'=x''+1\). We can rewrite the second DE as \(x''+1=2e^x+x'+x \implies x''-x'-x=2e^x-1\).

Answer 4

@imran, no you most definitely cannot do that because y is a function of x and vice versa so your integrations are incorrect.

@Mr.M: you've got it.

Answer 5

But e^x is a problem.

Answer 6

This is phase plane problem , btw

Answer 7

I don't even know what phase plane problems are!

Answer 8

b/c we can divide them
dx/dt=2y-x
dy/dt=e^x+y

dy/dx= e^x+y
-----
2y-x

(2y-x)dy= (e^x+y) dx
(e^x+y) dx +(-2y+x)dy= 0
exact

e^x+yx -y^x=C